Biological macromolecules can be imaged by single-particle cryogenic electron-microscopy (cryo-EM) without crystallization. This offers the possibility of imaging heterogeneous samples ? samples in which the macromolecule is in different conformational states. This research proposes a new method for analyzing such heterogeneous samples. Based on a Fourier-slice theorem for covariance functions, heterogeneity is analyzed by directly reconstructing 3d principal components and estimating the population densities of different conformations in the principal subspace, which is the subspace spanned by the principal components. This new methodology offers many advantages over the popular method of 3d classification: the new method works for continuous and discrete conformational states, it can organize conformational states in a meaningful way, and it can separate sample imperfections from conformational states. This research proposes to fully develop this methodology, validate it with simulations and real cryo-EM images, and release an open source software package for the use by the cryo-EM community.